Contact points for John's ellipsoid

Question

Suppose $K$ is a centrally symmetric convex body in $\mathbb{R}^n$ and $E$ is the John's ellipsoid, the ellipsoid of maximal volume inside $K$.

If $E$ and $K$ have exactly $2n$ contact points, say $(\pm x_i)_{i=1}^{n}$, do $(x_i)_{i=1}^n$ form an orthonormal basis for the Euclidean norm indiuced by $E$?

Naively, this statement seems true in two dimensions, but I don't know how to prove it. Or my intuition could be wrong.

Edit: removed another question (whether all points on $\partial K$ extreme points implies exactly $2n$ contact points) as it has an easy negative answer (in comments). Hopefully the remaining question is not as trivial.

Answer 1

Looks true. A necessary and sufficient condition for these points (let $E$ be a standard ball) is that the identity operator $I$ is a non-negative linear combination of projectors $P_i$ on lines through $x_i$: $$I=\sum c_i P_i.\quad\quad\quad\quad\quad(\heartsuit)$$ If $x_i$'s are linearly dependent, multiply $(\heartsuit)$ by a vector $y$ orthogonal to all $x_i$'s to get a contradiction. If not, denote by $(z_i)_{i=1}^n$ a biorthogonal system to $x_i$'s and multiply $(\heartsuit)$ by $z_j$ to get $z_j=c_jP_jz_j$. This means that $z_j$ is an eigenvector of $P_j$ with non-zero eigenvalue, thus $z_j$ is parallel to $x_j$. This is what you need.